Show that ` (a^2) /( x- alpha) +( b^2)/( x -beta) +(c^2)/( x- gamma)- x + delta =0` has only real roots if ` a ,b,c , alpha , beta , gamma , delta ` are all real .

If alpha , beta are the roots of x^2 + px +1=0 and gamma , delta are the roots of x^2 +qx +1=0 then (alpha - gamma )(beta - gamma)(alpha + delta )( beta + delta )=

If alpha , beta are the roots of x^2 +px -q=0 and gamma , delta are the roots of x^2 +px+r=0 then (alpha - gamma ) ( beta - gamma) ( alpha - delta ) ( beta- delta )=

If alpha , beta are the roots of x^2 - 3x +a =0 and gamma , delta are the roots of x^2 -12 x+b=0 and alpha , beta , gamma , delta in that order from a geometric progression increasing order with common ration r gt 1 then a+b=

If alpha, beta, gamma are the roots of x^(3) - 3x + 7 = 0 then alpha + beta + gamma =

If x = sin (alpha - beta) sin (gamma - delta) y = sin (beta - gamma) sin (alpha - delta ) and z = sin (gamma - alpha ) sin (beta - delta ) then

If alpha, beta, gamma are the roots of the equation x^(3) - 6x^(2) + 11x - 6 = 0 and if a = alpha^(2) + beta^(2) + gamma^(2) , b = alpha beta + beta gamma + gamma alpha and c = (alpha + beta) (beta + gamma)(gamma + alpha), then the correct inequality among the following is :

If alpha , beta are the roots of x^2+ax -b=0 and gamma , sigma are the roots of x^2 + ax +b=0 then (alpha - gamma )(beta - gamma )( alpha - sigma )( beta - sigma )=

If alpha , beta , gamma are the roots of x^3 -7x + 6 =0 the equation whose roots are alpha + beta , beta + gamma , gamma + alpha is

If (x + 2)(x + 3b) = c has roots alpha,beta , then the roots of (x + alpha)(x+beta) + c = 0 are